Chapter 13.5, Problem 70SR

To determine

To find : Exact value of $\sin \frac{7\pi }{8}$ .

Answer to Problem 70SR

  $0.382$

Explanation of Solution

Given information :

Expression $\sin \frac{7\pi }{8}$

Calculation :

The given equation is $\sin \frac{7\pi }{8}$

We can write $\sin \left(\pi -\frac{\pi }{8}\right)=\sin \left(\frac{\pi }{8}\right)$

Formula, $\cos 2a=1-2{\sin }^{2}a$

  $\begin{array}{l}\Rightarrow 2{\sin }^{2}a=1-\cos 2a\\ Fora=\frac{\pi }{8}\\ \Rightarrow 2{\sin }^2\frac{\pi }{8}=1-\cos 2\left(\frac{\pi }{8}\right)=1-\cos \frac{\pi }{4}=1-\frac{1}{\sqrt{2}}=\frac{\sqrt{2}-1}{\sqrt{2}}\\ \Rightarrow {\sin }^2\frac{\pi }{8}=\frac{\sqrt{2}-1}{2\sqrt{2}}=\frac{\sqrt{2}-1}{2\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}(\sqrt{2}-1)}{4}\\ \Rightarrow {\sin }^2\frac{\pi }{8}=\frac{2-\sqrt{2}}{4}\\ \Rightarrow \sin \frac{\pi }{8}=\frac{\sqrt{2-\sqrt{2}}}{2}\text{(Ignore negative value)}\\ \Rightarrow \sin \frac{7\pi }{8}=\sin \frac{\pi }{8}=\frac{\sqrt{2-\sqrt{2}}}{2}=\frac{0.765}{2}=0.382\end{array}$

  $\begin{array}{l}\text{The given equation is }\sin \frac{7\pi }{8}\\ Wecanwrite\sin \left(\pi -\frac{\pi }{8}\right)=\sin \left(\frac{\pi }{8}\right)\\ \text{Formula, }\cos 2a=1-2{\sin }^{2}a\\ \Rightarrow 2{\sin }^{2}a=1-\cos 2a\\ Fora=\frac{\pi }{8}\\ \Rightarrow 2{\sin }^2\frac{\pi }{8}=1-\cos 2\left(\frac{\pi }{8}\right)=1-\cos \frac{\pi }{4}=1-\frac{1}{\sqrt{2}}=\frac{\sqrt{2}-1}{\sqrt{2}}\\ \Rightarrow {\sin }^2\frac{\pi }{8}=\frac{\sqrt{2}-1}{2\sqrt{2}}=\frac{\sqrt{2}-1}{2\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}(\sqrt{2}-1)}{4}\\ \Rightarrow {\sin }^2\frac{\pi }{8}=\frac{2-\sqrt{2}}{4}\\ \Rightarrow \sin \frac{\pi }{8}=\frac{\sqrt{2-\sqrt{2}}}{2}\text{(Ignore negative value)}\\ \Rightarrow \sin \frac{7\pi }{8}=\sin \frac{\pi }{8}=\frac{\sqrt{2-\sqrt{2}}}{2}=\frac{0.765}{2}=0.382\end{array}$